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ELE2103

Linear systems and controlFaculty of Engineering and Surveying

In t roduc to ry bookSemester 2 2011

Published by

University of Southern QueenslandToowoomba Queensland 4350Australia

http://www.usq.edu.au

© University of Southern Queensland, 2011.2.

Copyrighted materials reproduced herein are used under the provisions of the Copyright Act 1968 as amended, or as a result of application to the copyright owner.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without prior permission.

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Table of contents

Page

Essential information 1

Introduction 2Course overview 3

Linear Systems and control 3

Concept map 4

Study materials 5

Study schedule 6

Assessment 7

Late assignment policy 7

Assignment 1 8System modeling, time response and stability 8

Assignment 2 10Block diagram, system performance and responses 10

Past examinations 11

ELE2103 – Linear systems and control 1

Essential information

The topics in the following list provide important information that will assist you with your study. You can access a handout containing the information on your StudyDesk through the ‘Essential information (study materials)’ link <http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/essentialhandout.pdf>. You will need your UConnect username and password to access the file. Please make sure you read this information carefully before commencing your study.

• Getting started <http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/getting_started.pdf>

• Course specification <http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/course_specification.pdf>

• Support <http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/support.pdf>

• UConnect<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/u_connect.pdf>

• Assignment submission<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/assignment_submission.pdf>

• Grading levels<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/grading_levels.pdf>

• Course evaluation <http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/course_evaluation.pdf>

• Residential schools<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/residential_school.pdf>

• Library<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/library.pdf>

• Referencing APA<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/apa_referencing_guide.pdf>

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• Optional purchase of study materials<http://usqstudydesk.usq.edu.au/file.php/1/sitefiles/DeC/essential_info/optional_purchase.pdf>

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2 ELE2103 – Linear systems and control

Introduction

This course is an amalgamation of two smaller courses in the previous BEng course, viz.

Systems Modelling and Analysis and Control Engineering 1.

• Since the first course was very much a preparation for the second (as well as other areas, of course) it is appropriate to combine the two into a single package. The content of the course and the rationale for it are introduced in module 1, so very little needs to be said here.


Course overview

Linear Systems and control

Fourier series

Introduction

Linear Modelling

Root Locus Concept

Laplace Transform

Modelling

ODE Description

Transfer Functions

Design via Bode diagram

Design via Root Locus

Response in Freq. Domain

Fourier Transforms

Frequency response;

Bode diagrams

Block diagrams;

PFs; poles; zeros;

1st, 2nd, high order

Performance improvement;

PID; Phase lead; lag.

Stability; Nyquist & Theorem,

GM, �� Routh Hurwitz

Signals Systems

Convolution – system

Steady state performance;

Error co-efficients

response in time domain


Concept map

Linear

Lumped Parameter

Time Invariant Systems

Described By

Ordinary Differential Equations

Test Signals

Impulse

Step

Ramp

Parabolic

Periodic

Signals

Laplace

Transform

Fourier

Analysis

LAPLACE TRANSFORM

TRANSFER FUNCTIONL

Frequency Domain Behaviour

Inverse

Laplace

Transform

Stability and Behaviour of

Poles and Zeros

Transient

Response

DelaysFirst Order

Systems

Second Order

Systems

Compensation

Routh-Hurwitz

Criterion

Root

Locus

Bode

Plots

Nyquist

Plots

Steady

State Errors

Phase & Gain

Margin

Higher Order

Systems


Study materials

Introductory book

References:

Study book: Linear Systems and Control

Textbook: Nise, NS 2008, Control system engineering, 5th edn, John Wiley & Sons, New York.

Support material: Matlab student edition (version 6.0/later and Control Toolbox)

1. Kuo, BC & Golnaraghi, F 2010, Automatic control systems, 9th edn, John Wiley & Sons.

2. Katsuhiko, Ogata 2009, Modern control engineering, 5th edn, Prentice Hall, Inc.

3. Dorf, RC & Bishop, RH 2008, Modern control systems, 11th edn, Prentice Hall.


Study schedule

Week Module Activity/Reading Assessment

118–22 July

1. Linear Systems Analysis

Study Book: Module 1Textbook: Chapter 1

225–29 July

2. Laplace Transforms and Transfer Function


31–5 August

3. System Concepts Study Book: Module 3Textbook: Chapter 3+5

48–12 August

4. Time Response Study Book: Module 4+8Textbook: Chapter 4+6

Reminder: End of week 4 is the last date to drop S2 courses without academic or financial penalty.

515–19 August

5. The Root Locus Study Book: Module 5Textbook: Chapter 8

622–26 August

6. Fourier Transform Study Book: Module 6

729 Aug–2 Sept

7. Frequency Response


Assignment 1Due: 2 Sep. 2011

85–9 September

8. Stability of Control Systems

Study Book: Module 8

Reminder: End of week 8 is the last date to drop S2 courses without academic penalty.

912–16 September

BREAK

1019–23 September

BREAK

1126–30 September

9. Control System Characteristics


123–7 October

10. Performance Improvement

Study Book: Module 10Textbook: Chapter 9+11

1310–14 October

11. Controller Design Study Book: Module 11Textbook: Chapter 9+11

1417–21 October

11. Controller Design Study Book: Module 11Textbook: Chapter 9+11

Assignment 2Due: 21 Oct. 2011

1524–28 October

Revision

16–1731 Oct–11 Nov

EXAMINATION PERIOD


Assessment

Assessment of this course is via two assignments and an examination; please note the dates for these. A study chart is provided in this book and is offered as a guide with which to check your progress – it might be worth looking at it now and again.

The assessment referred to above is for the purposes of academic administration. The real assessment will come from yourself – if you can use this material appropriately to describe signals and systems, to make valid measurements of their behaviour and, perhaps, to invent and implement a first rate control system then the course will have been a success.

Exam details

The final assessment for this course will be a restricted examination for 700 of the total 1 000 marks for the course. The format of this examination is as follows:

Late assignment policy

Please consult the course specification for up to date information regarding late assignments.

Please also note that assignment extensions are unlikely to be given and that late assignments will be penalised at a rate of 5% per day late.

Perusal – 10 minutes

Duration – 2 hours

This examination assesses all study material.

Drawing, writing implements, and a non programmable calculator are allowed into the exam.

Study materials are not allowed in to the exam venue.

Assessment details

Assignment Due Weighting (%)

Assignment 1Assignment 2Examination (restricted)

2 September 201121 October 2011End of semester

201070


Assignment 1

System modeling, time response and stability

Physical system modeling, time response and stability

For a motor, load and torque-speed system shown below

1. Find the differential equation relating input ea(t) and output L(t). (Hint:

where ea(t) and ia(t) are the armature voltage and current, Jm and Dm are the equivalent inertia and damping; Tm(t) is the torque developed by the motor; N1/N2 is the ratio of the gear system.)

(50 marks)

2. Determine the system transfer function G(s)= L(s)/Ea(s). (30 marks)

3. Given Ja=5, JL=700, Da=2, DL=800, N1=100, N2=1000, Ka=5, Kb=2 and Ra=1, obtain the time response for a unit impulse input.

(30 marks)

Due date: 2 September 2011Value: 20%Total marks: 200Penalty for late submission: 5% per day

θL(t)

θm(tea(t)

(t)

θ

)()( ;)(

)()(

)()( ; ;1

2

2

2

1

2

2

1

tiKtT

dt

tdKtiRte

t

N

NtD

N

NDDJ

N

NJJ

aam

m

baaa

LmLamLam

θ


4. If the above system is considered as a plant, and a feedback control system is formed as below, for gain K=40, obtain the unit step response of the feedback control system.

(40 marks)

5. Draw a root locus and analyze the stability of the feedback system in 4).(50 marks)

ea(t) L(t) r(t)

Plant + K


Assignment 2

Block diagram, system performance and responses

1. Simplify the above block diagram and determine the closed-loop transfer function (Hints: Join summing junction 2 and 3, then simplify the block diagram using parallel, cascade and feedback connections)

(15 marks)

2. If given transfer function and H(s)=1, using Routh-

Hurwitz criterion, find the value of K that will yield oscillations for this unity feedback system.

(20 marks)

3. Given K=307.8 in question 2), draw its Bode plots and determine the gain margin and phase margin for the system).

(50 marks)

4. Given K=307.8 in question 2) again, find the steady-state errors for a unit step input and a unit ramp input respectively.

(10 marks)

5. If you are required to improve the steady state error and make it to be zero for a unit step input, should you choose a PI or PD controller?

(5 marks)

Due date: 21 October 2011Value: 10%Total marks: 100Penalty for late submission: 5% per day

G s( ) 100Ks 15+( ) s 27+( ) s 39+( )

-----------------------------------------------------------=


Past examinations

STUDENT NAME: STUDENT NO.:

UNIVERSITY OF SOUTHERN QUEENSLAND

FACULTY OF ENGINEERING AND SURVEYING

Course No: ELE2103 Course Name: LINEAR SYSTEMS AND CONTROL

Assessment No: 3 Internal

External

This examination carries 70% of the total

assessment for this courseX

X

Examiner: PAUL WEN Moderator: MARK PHYTHIAN

Examination Date: NOVEMBER 2009

Time Allowed: Perusal – Ten (10) minutes

Working – Two (2) hours

Special Instructions:

This is a RESTRICTED examination.

Writing materials and non-programmable calculators are permitted. Students must note the maker and mode

of the calculator used on the front of the answer book (or examination paper where applicable). This may be

subject to checking by the examination supervisor.

Students are permitted to write on the examination paper during perusal time.

Students must complete questions 1 to 3, and question 4 or 5. Questions 4 and 5 are alternatives. One

and only one will be marked.

All examination question papers must be submitted to supervisors at the end of every examination and

returned to USQ.

Any non-USQ copyright material used herein is reproduced under the provisions of Section 200(1)(b) of the

Copyright Amendment Act 1980.

ELE2103 - LINEAR SYSTEMS AND CONTROL NOVEMBER 2009 Page 1

QUESTIONS 1 TO 3 ARE COMPULSORY

QUESTION 1 (100 marks total)

Briefly describe each of the following terms in the context of linear systems and control. Describe

mathematically, if possible. Do not write more than three dot points per term. (10 marks each term)

1. Open-loop system transfer function

2. Poles and zeros

3. Transient response

4. Steady state error

5. Root locus

6. Bode plots

7. PID controller

8. Phase lead compensator

9. Overshoot

10. Settling time

Page 1 of 9

ELE2103 - LINEAR SYSTEMS AND CONTROL NOVEMBER, 2009 Page 2

Page 2 of 9


For a mass-spring-damper mechanical system shown below,

Part 1 70 Marks

Verify the following differential equations relating input force f(t) and output displacement x1(t) and x2(t) in the

above system (Hint: K, fv and M are spring constant, friction coefficient and mass respectively)

2

1

2

11

11112212

2

2

2

2

212212

22

3

)()()())()((

))()((

)())()((

))()(()()(

dt

txdM

dt

tdxftxKtxtxK

dt

txtxdf

dt

txdMtxtxK

dt

txtxdf

dt

tdxftf

vv

vv

Part 2 50 Marks

Determine the system transfer function G(s)= X1(s)/F(s)

Part 3 80 Marks

For this part, suppose system transfer function)49.12)(4.1(

3.13)(

2sss

ssG

Obtain the system impulse response in time domain.


Page 3 of 9


For a position control system shown in the figure below,

Part 1 50 Marks

Determine the open-loop transfer function G(s)H(s) and the closed-loop transfer function T(s).

Part 2 50 Marks

Find the unit step steady state error of the above system.

Part 3 100 Marks

Design K and K2 in the system to yield an overshoot of 16% and a settling time of 0.8 second.


Page 4 of 9

COMPLETE EITHER QUESTION 4 OR QUESTION 5

QUESTION 4 (200 marks total, alternative of Question 5)

A floppy disk driver is a position control system in which a read/write head is positioned over a magnetic disk.

The system responds to a command from a computer to position itself at a particular track on the disk. A physical

representation of the system and a block diagram are shown in the figure below

where K is a gain which can be adjusted according to the requirements. Assume K is 1 in your Bode plots

drawing.

Part 1 120 Marks

Sketch the Bode plots for the above system using straight-line approximation, for from 0.1 to 1000

rad/s. Show the sketch procedure. (A graph paper is provided at the end)

Part 2 60 Marks

Estimate the phase and gain margins based on your Bode plots.

Part 3 20 Marks

If you are required to improve system transient response such as overshot and settling time, what simple

compensator/controller should you include in the forward path of the system?


Page 5 of 9


The open-loop transfer function of a unit feedback control system is given as

1)2(

4)()(

ss

sKsHsG

p

where Kp is a proportional controller which can be adjusted according to the system design specification.

Part 1 120 Marks

Draw the root locus diagram of the system for 0<Kp<+ and show the drawing steps.

Part 2 60 Marks

Determine the range of Kp for a stable system using Routh-Hurwitz criterion.

Part 3 20 Marks

Briefly explain your conclusion obtained in part 2 using your root locus diagram in part 1.

END OF QUESTIONS

THE FOLLOWINGS ARE EXAMINATION INFORMATION SHEETS


Page 6 of 9

Selected Laplace Transforms

f(t) F(s)

)(t , Impulse 1

u(t), step s

1

tn

1

!n

s

n

e-at

as

1

te-at

2)(

1

as

)(1 btat

abee

))((

1

bsas

e-at

sin t22)( as

e-at

cos t22)( as

as

Selected Laplace Transform Theorems

Theorem Name

0)()]([)( dtetftfLsF

st Definition

)()()]()([ 212211 sFksFktfktfkL Linearity)()]([ asFtfeL

at

Frequency Shift )()]([ sFeTtfL

st

Time Shift

)(1

)]([a

sF

a

atfL Scaling

)0()()(

fssF

dt

tdfL Differentiation

n

k

kknn

n

n

fssFs

dt

tfdL

1

1 )0()()( Differentiation

s

sFdfL

t )()(

0

Integration

)(lim)(0

ssFfs

Final Value )(lim)0( ssFf

sInitial Value


Page 7 of 9

Selected Formulae

First order system, its step response & specification

aa

T

a

Tetc

assass

asRsGsC

as

asG

sr

at1

;4

;2.2

;1)(

11

)()()()(;)(

Second order system, its step response & specification

)100/(%ln

)100/ln(%;100%;

1;

4

1tan;1cos

1

11

1sin1

1cos1)(

)2()()()(;

2)(

22

1

2

2

12

2

2

2

2

22

2

22

2

2

OS

OSeOSTT

te

ttetc

sss

sRsGsC

ss

sG

n

p

n

s

n

t

nn

t

nn

n

nn

n

n

n

Mason’s rule

k

kkT

sR

sCsG

)(

)()(

Routh-Hurwitz stability criterion

Steady-state error for unit feedback system and specification

)(lim);(lim);(lim

)](1)[(lim)(lim)(lim)(

2

000

00

sGsKssGKsGK

sTssRssEtee

sa

sv

sp

sst


Page 8 of 9

Root locus

2,1,0;##

)12(;

##

180)12()()(&1)()( 0

k

zerospoles

k

zerospoles

zerospoles

ksHsKGsHsKG

aa

Controllers

C

C

CCPD

C

PI

PID

ps

zsKsGzsKsG

s

zsKsG

s

bassK

s

sKKsKsK

s

KKsG

)();()(;)(

)(22

3213

21

Frequency response, Bode plots and Nyquist plot

2

2

222222

2

22

2

0

22

2

0

22

2

2222

022

2

n

022

2

2222

0

0

0

0

21;12

1

4)(|)(|

2)(

40log20logMor18011

)(

;01

20log20logMor011

)(

12

11

2

1)(

40log20logMor180)(

;020log20logMor0)(

122)(

20log20logMor901

)(

;020log20logMor011

)(

1

111)(

20log20logMor90)(

;020log20logMor0)(

1)(

npp

nn

n

nn

n

nnn

nn

nnn

nn

nn

nnn

M

jGM

ss

sGFor

whilejG

whilejG

ssss

sGFor

whilejG

whilejG

sssssGFor

whilej

jG

whilea

aa

jG

a

saas

sGFor

whilejjG

whileaaajG

a

saassGFor


Page 9 of 9

You can rescale the axes to fit your plots

10-1

100

101

102

103

104

-200

-150

-100

-50

0

50

10-1

100

101

102

103

104

-200

-150

-100

-50

0

50

Any non-USQ copyright material used herein is reproduced under the provisions of Section 200(1)(b) of the

Copyright Amendment Act 1980.

STUDENT NAME: STUDENT NO.:

UNIVERSITY OF SOUTHERN QUEENSLAND

FACULTY OF ENGINEERING AND SURVEYING

Course No: ELE2103 Course Name: LINEAR SYSTEMS AND CONTROL

Assessment No: 3 Internal

External

This examination carries 70% of the total

assessment for this course

Examiner: PAUL WEN Moderator: MARK PHYTHIAN

Examination Date: NOVEMBER 2010

Time Allowed: Perusal – Ten (10) minutes

Working – Two (2) hours

Special Instructions:

This is a RESTRICTED examination.

Writing materials and non-programmable calculators are permitted. Students must note the maker and mode

of the calculator used on the front of the answer book (or examination paper where applicable). This may be

subject to checking by the examination supervisor.

Students are permitted to write on the examination paper during perusal time.

Students must complete questions 1 to 3, and question 4 or 5. Questions 4 and 5 are alternatives. One

and only one will be marked.

All examination question papers must be submitted to supervisors at the end of every examination and

returned to USQ.

X

X


Page 1 of 9

QUESTIONS 1 TO 3 ARE COMPULSORY


Briefly describe each of the following terms in the context of linear systems and control. Describe

mathematically, if possible. Do not write more than three dot points per term. (10 marks each term)

1. System transfer function

2. Closed-loop system

3. Steady state response

4. Damping factor

5. Frequency response

6. Magnitude plot

7. Peak time

8. Settling time

9. Phase lag compensator

10. PID controller


Page 2 of 9


Consider a leaky tank system shown below. The inflow is u(t). The outflow is v(t) = ch(t), where c is a

constant coefficient. The volume is given by ah(t), where a is the cross sectional area of the tank. The

net volume gain is given by (inflow – outflow).

Part 1 60 Marks

Write a differential equation modelling the relationship between u(t) and v(t). Then find out its transfer

function )(

)(

sU

sV.

Part 2 40 Marks

The outflow v(t) of the above tank is the inflow to another tank. The second tank has a cross sectional

area of a2, a coefficient c2, and an outflow y(t). For this double-tank system, find out its transfer function

between input U(s) and output Y(s).

Part 3 100 Marks

Given a=c=a2=1 and c2=2, derive the unit step response of the double-tank system.

(Assume )2)(1(

2

)(

)(

sssU

sY if part 2 failed)

Inflow u(t)

h(t)

outflow v(t)=ch(t)


Page 3 of 9


For the system shown in the figure below,

Part 1 60 Marks

Determine the open-loop transfer function G(s) and the closed-loop transfer function T(s). (Hints: simplify the

block diagram by shifting the 2nd pick off point to the 3rd)

For the following parts, assume the open-loop transfer function )(

1)()()(ass

KsGsHsG

,

Part 2 40 Marks

Determine the type of the closed-loop system and the steady state error for a unit step input.

Part 3 100 Marks

Determine the value of K and a, so that the percent overshoot %OS of the closed-loop system for a unit step input

is limited to 16.303% and the settling time Ts is 1.6 seconds.


Page 4 of 9

COMPLETE EITHER QUESTION 4 OR QUESTION 5


The open-loop transfer function of a control system is )6415(

)(2

sss

KsG

Part 1 120 Marks

For K=128 sketch the Bode plots for the above system using straight-line approximation.

Part 2 40 Marks

Estimate the phase margin and gain margin based on the Bode plots.

Part 3 40 Marks

Is the closed-loop system stable? Explain your conclusion.


Page 5 of 9


The open-loop transfer function of a unit feedback control system is given as

)3)(2)(1()(

sss

KsG

p

where Kp is a proportional controller which can be adjusted according to the requirements.

Part 1 100 Marks

Sketch the root locus diagram of the system and show the drawing steps.

Part 2 40 Marks

Based on the root locus obtained in part 1, discuss the system stability when Kp is adjusted from 0 to .

Part 3 60 Marks

Determine the range of Kp for a stable system using Routh-Hurwitz criterion. Does the result agree with your

discussion in part 2?

END OF QUESTIONS

THE FOLLOWINGS ARE EXAMINATION INFORMATION SHEETS


Page 6 of 9

Selected Laplace Transforms

f(t) F(s) )(t , Impulse 1

u(t), step s

1

tn 1

!ns

n

e-at

as

1

te-at

2)(

1

as

)(1 btat

abee

))((

1

bsas

e-atsint 22)(

as

e-atcost 22)(

as

as

Selected Laplace Transform Theorems

Theorem Name

0

)()]([)( dtetftfLsF st Definition

)()()]()([ 212211 sFksFktfktfkL Linearity )()]([ asFtfeL at Frequency Shift )()]([ sFeTtfL st Time Shift

)(1

)]([a

sF

aatfL Scaling

)0()()(

fssF

dt

tdfL Differentiation

n

k

kknn

n

n

fssFsdt

tfdL

1

1 )0()()( Differentiation

s

sFdfL

t )()(

0

Integration

)(lim)(0

ssFfs

Final Value )(lim)0( ssFf

s Initial Value


Page 7 of 9

Selected Formulae

First order system, its step response & specification

aaT

aTetc

assass

asRsGsC

as

asG

sr

at 1 ;

4 ;

2.2 ;1)(

11

)()()()( ;)(

Second order system, its step response & specification

)100/(%ln

)100/ln(% ;100% ;

1 ;

4

1tan ;1cos

1

11

1sin1

1cos1)(

)2()()()( ;

2)(

22

1

2

2

12

2

2

2

2

22

2

22

2

2

OS

OSeOSTT

te

ttetc

ssssRsGsC

sssG

n

p

n

s

n

t

nn

t

nn

n

nn

n

n

n

Mason’s rule

k

kkT

sR

sCsG

)(

)()(

Routh-Hurwitz stability criterion

Steady-state error for unit feedback system and specification

)(lim );(lim );(lim

)](1)[(lim)(lim)(lim)(

2

000

00

sGsKssGKsGK

sTssRssEtee

sa

sv

sp

sst


Page 8 of 9

Root locus

2,1,0 ;

##

)12( ;

##

180)12()()( & 1)()( 0

kzerospoles

k

zerospoles

zerospoles

ksHsKGsHsKG

aa

Controllers

C

CCCPD

CPI

PID

ps

zsKsGzsKsG

s

zsKsG

s

bassK

s

sKKsKsK

s

KKsG

)( );()( ;)(

)(22

3213

21

Frequency response, Bode plots and Nyquist plot

2

2

222222

2

22

2

0

22

2

0

22

2

2222

022

2

n

022

2

2222

0

0

0

0

21 ;12

1

4)(|)(|

2)(

40log20logMor 18011

)(

;0 1

20log20logMor 011

)(

12

11

2

1)(

40log20logMor 180)(

;0 20log20logMor 0)(

122)(

20log20logMor 901

)(

;0 20log20logMor 011

)(

1

111)(

20log20logMor 90)(

;0 20log20logMor 0)(

1)(

npp

nn

n

nn

n

nnn

nn

nnn

nn

nn

nnn

M

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You can rescale the axes to fit your plots

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ele2103 linear systems and control - transtutors · ele2103 – linear systems and control 9 4. if...

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